ON A MIXED-HYBRID FINITE ELEMENT METHOD FOR ANISOTROPIC PLATE BENDING PROBLEMS

The paper deals with a new mixed-hybrid finite element method for anisotropic elastic plate bending problems, which gives a simultaneous approximation to bending and twisting moment tensor and deflection fields in the interior of each triangle and traces of the displacement field and its normal derivative on sides of each triangle. Efficient computer implementation procedures for this mixed-hybrid scheme have been developed. Results of numerical experiments on different anisotropic plates of practical and research interest are given.     

Analytical Solution for an Orthotropic Elastic Plate Containing Cracks

The problem of estimating the bending stress distribution in the neighborhood of a crack located on a single line in an orthotropic elastic plate of constant thickness subjected to bending moment or twisting moment is examined. Using classical plate theory and integral transform techniques, the general formulae for the bending moment and twisting moment in an elastic plate containing cracks located on a single line are derived. The solution is obtained in a closed form for the case in which there is a single crack in an infinite plate and the results are compared with those obtained from the literature.     

DSC analysis of a simply supported anisotropic rectangular plate

The discrete singular convolution (DSC) algorithm is used to analyze the deflection and free vibration behavior of a simply supported anisotropic rectangular plate. A novel approach is proposed to solve the difficulty in using DSC to handle the simply supported boundary conditions with bending–twisting coupling. DSC results are presented for bending under distributed load and a center concentrated load, and natural frequencies of flexural vibrations. It is shown that the DSC with proposed method to apply the simply supported boundary conditions yields very accurate results as compared to exact solutions or results obtained by methods of differential quadrature and finite element with fine meshes. It is also verified that neglecting the bending–twisting coupling in applying the simply supported boundary conditions may result incorrect solutions, especially for the bending analysis of anisotropic plates.     

A mixed problem of plate bending for a regular octagon weakened with a required full-strength hole

The paper addresses a problem of bending of an isotropic elastic plate, bounded by a regular octagon weakened with a required full-strength hole including the origin of coordinates. Coordinate axes pass the middle points of opposite parallel sides of octagon. Rigid bars are attached to each component of the broken line of the outer boundary of the plate. This plate bends under the action of concentrated moments applied to the middle points of the bars. Unknown part of the boundary is free from external forces. Using the methods of complex analysis, the plate deflection and required full-strength contours are determined. Numerical analysis is performed, and the corresponding graphs are constructed.     

The general solutions of an infinite plate with doubly periodic cracks under bending and twisting

The objective of the present paper is to use the basic theorem of Jacobi elliptic functions and residues to find the general solutions of the stress intensity factor of a plate with doubly periodic cracks subjected to a pair of concentrated couples $\text{M}{}_{\mathrm{b}}$ and $\text{M}{}_{\mathrm{t}}$ on the faces of each crack. In this paper the general solutions are expressed in simply closed forms, and they can also be applied to solve practical problems of bending and torsion of a rectangular sheet containing a central crack. The general solutions may be used as a fundamental Green's function for finding other solutions of a plate with doubly periodic cracks under arbitrary bending and twisting moments. It is easily proved that all problems of bending and torsion of an infinite plate with a central crack or singly periodic cracks, are the special cases of the general solutions in this paper.     

Mixed-hybrid scheme of the finite element method for solving the problems on bending,free vibration,and stability of plates

To solve a problem on bending, vibration, and stability of plates, a hybrid finite element has been constructed on the basis of Zienkiewicz’s triangle. A mixed approximation is used for the plate deflection and turning angles. It is shown that with a decrease in the triangle dimensions the mixed approach ensures convergence both for the plate deflection and the bending moments, which is practically independent of the way the plate is split into triangular elements. In the problems on free vibrations and stability of plates, the mixed approach yields more exact values of the eigenfrequencies and critical loads as compared to a classical Zienkiewicz’s triangle. The results of the numerical analysis of the convergence and accuracy of the solutions to a number of test problems on bending, free vibration, and stability of a square plate are presented. __________ Translated from Problemy Prochnosti, No. 4, pp. 108–122, July–August, 2008.     

层合板中压电作动器最优厚度和嵌入位置研究

研究了压电结构中压电片厚度和嵌入深度的优化问题。首先给出了压电层合板的高阶耦合分析模型;然后以不受约束的含压电铺层复合材料板为代表,在压电层厚度方向施加电场时板自由变形,假设板任意微元横截面上内力为零,以其弯(扭)曲曲率最大为优化目标,建立了求解压电片最优厚度和嵌入深度问题的约束优化模型。最后分别以各向同性板中嵌入各项同性压电片和复合材料板中嵌入各向异性压电片为例进行了分析,绘出了目标函数的三维曲面图及等高线图,结果表明压电片的作动效能与其厚度和嵌入位置密切相关,而最优厚度和嵌入位置是由压电片和基体的材料特性决定的。     

Plate bending for frame analysis by coupling method of singular integral operators method with finite elements

The coupling of the Singular Integral Operators Method (SIOM) and the Finite Element Method (FEM) is proposed and investigated for the solution of plate bending problems. Such combination of the above two numerical methods gives more accurate results for the solution of plate bending problems in structural analysis. The plates are basic units of building frame structures, as well as to underground structures and thus the determination of bending moments around the plates is very important for the analysis of all kinds of structures. An application of plate bending problems is finally given to the determination of the bending moments in a rectangular plate with a rectangular hole in the center of the plate.     

A finite element application of the modified Rayleigh–Ritz method

Considerable attention has been devoted in the literature on numerical methods towards securing energy convergence of solutions for, say, linearly elastic plate bending problems. Although energy convergence is necessary it by no means follows that the derived bending moments and shearing forces converge uniformly at a given point and it is this kind of feature which the engineer is really seeking. This question is examined in the context of a problem which is of particular interest to the civil engineering field and concerns the bending of a square plate under uniformly distributed load; the plate has simply supported edges and contains a central square hole with free edges. The solution to this multiply connected and mixed boundary value problem is obtained through a recently developed modification to the Rayleigh–Ritz method which has very general application and renders the solution mathematically valid up to the internal corner points where the bending moments are singular. Use is made of triangular equilibrium finite elements in conjunction with continuous eigenfunctions. Although it is already known that the order (i.e. the eigenvalue) of the singularity at the internal corners is available by inspection, it is an interesting feature of the present solution that a good approximation to the amplitude is also obtained by an inspection of the finite element results.     

Identities for the fundamental solution of thin plate bending problems and the nonuniqueness of the hypersingular BIE solution for multi-connected domains

Four integral identities for the fundamental solution of thin plate bending problems are presented in this paper. These identities can be derived by imposing rigid-body translation and rotation solutions to the two direct boundary integral equations (BIEs) for plate bending problems, or by integrating directly the governing equation for the fundamental solution. These integral identities can be used to develop weakly-singular and nonsingular forms of the BIEs for plate bending problems. They can also be employed to show the nonuniqueness of the solution of the hypersingular BIE for plates on multi-connected (or multiply-connected) domains. This nonuniqueness is shown for the first time in this paper. It is shown that the solution of the singular (deflection) BIE is unique, while the hypersingular (rotation) BIE can admit an arbitrary rigid-body translation term in the deflection solution, on the edge of a hole. However, since both the singular and hypersingular BIEs are required in solving a plate bending problem using the boundary element method (BEM), the BEM solution is always unique on edges of holes in plates on multi-connected domains. Numerical examples of plates with holes are presented to show the correctness and effectiveness of the BEM for multi-connected domain problems.     

Resonant fatigue test bench for shaft testing

Shaft fatigue testing involves long test times (~3 months), high energy consumption and high test equipment maintenance costs owing to the high bending and twisting moments required (~1600 Nm), high number of cycles (~10⁷) and large sample sizes (~30). To reduce testing time, we designed, manufactured and evaluated a resonant plate test bench. Using finite element analysis and topological optimization, we redesigned the traditional resonant flat plates for higher resonant frequency and lower deflection at the plate free ends. We found that the optimal topology is an I‐beam; it doubles the frequency of flexion tests, up to 100 Hz, and exhibits 2 mm of deflection under a load of 1 kN. To measure the moments induced on the shaft sample during testing, we measured the surface deflection of the resonant plates. Tests on a calibration axle showed that the induced shaft moment has very high linear correlation (R² > 0.99) with the plate's surface deformation. We used our test bench to evaluate fatigue resistance for a type of crankshaft manufactured by a local company. We found that their fatigue resistance limit was 1250 Nm at 10⁷ cycles and that their mean useful life was 10⁴ cycles when they are subjected to a 1400 Nm moment. These results agree with previous results on this type of crankshaft using other methods.     

Revisit of an elliptic hole problem by using complex variable method

In this paper, the problem of an elliptic hole embedded in an infinite plate interacting with an arbitrary point load is revisited by using the complex variable method. Based on analytical continuation theorem, the continuity conditions across the interface are automatically satisfied in a straightforward manner. It is shown that the solution for an infinite domain with an elliptic hole can be obtained from the solution of the corresponding homogeneous problem merely by a simple algebraic expression. This relation is universal in the sense of being independent of the loading considered. The solution of the corresponding homogeneous problem is considered as the principal part of the complex potentials while the complementary part of the complex potentials can be obtained by using analytical continuation theorem. Different expressions of the complementary part of the complex potentials are presented in this paper which are all proved to be the same result.     

Failure prediction of a plate weakened by an elliptic hole under thermal or mechanical load

The failure initiation and crack trajectory are predicted for a plate weakened by an elliptic hole under thermal or mechanical load. The closed form solutions are obtained by using the complex variable theory and the fracture behavior is predicted by applying the strain energy density criterion. The results lead to a number of important conclusions. A general trend is that failure initiation always occurs at the site near the interface with material inhomogeneity while the fracture trajectory tends to spread away from the boundary of an elliptic hole whether thermal or mechanical loads are imposed in the given problem. Fracture initiation is most likely to take place at the site with relatively low temperature accompanied by the presence of tensile stress. Depending on material geometry, kinking phenomena are observed at the boundary layer adjacent to the interface. In general, the critical load in compression is larger than that in tension.     

Simply supported plates by the boundary integral equation method

Plates governed by Kirchhoff's equation have been analysed by the boundary integral equation method using the fundamental solution of the biharmonic equation. In the case of supported plates, the boundary conditions permit the uncoupling of the field equation into two harmonic equations that originate, due to the nature of the fundamental solution, easier integration kernels and a simpler system of equations. The calculation of bending and twisting moments and transverse shear force can be formed, combining derivatives of the integral equation which defines the expression of the deflection on any point of the plate. The uncoupling of the biharmonic equation into two Poisson's equations involves the discretization of the domain of the studied problems. Nevertheless, the unknown quantity of the problem does not appear in the domain integrations for which a refined discretization is unnecessary. In the paper, however, a numerical alternative is considered to express the domain integral by means of boundary integrals. In this way, we need only discretize the boundary of the plate, making it necessary to solve a supplementary system of equations in order to calculate the coefficients of the approximation carried out.     

Three-dimensional finite element for analysing thin plate/shell structures

A three-dimensional (3-D) hexahedron finite element is presented for the analysis of thin plate/shell structures. The element employs an explicit algebraic definition of six uniform (continuum) strains, six rigid body modes and classical Lagrange-Germain-Kirchhoff thin plate bending modes. Nine additional stiffness factors are used to control higher-order hourglass modes. The element may be used for plate/shell analyses where the flat plate assumptions are appropriate. Also it can easily be adapted to form transition elements to lower order 2-D elements, or to higher-order 3-D continuum elements. The stiffness matrix satisfies the geometric isotropy requirement, passes the patch test, and gives essentially identical response to either applied transverse corner forces or to twisting moments applied on the corner, a requirement of Kirchhoff's corner conditions for a classical thin plate. Several examples are presented to demonstrate the performance of this finite element.     

Thin plate bending of dissimilar half-planes with interface debonding emanating from an elliptical hole

The problem of thin plate bending of two bonded dissimilar half-planes containing an elliptical hole on the interface with debonding emanating on both sides is presented. The external load is a uniformly distributed bending moment applied at infinity perpendicular to the interface. An analytical solution is obtained using the complex stress function approach and the rational mapping function technique. Stress distributions on the interface, the boundaries of the hole and in the vicinity of the debonded tips are obtained. The stress intensity of debonding is obtained for aribitrary lengths of debonding, all possible hole dimensions and rigidity ratios.     

Transverse shear and pressurization effects in the bending of plates with reinforced cracks

The authors develop an eigth-order model for bending of transversally isotropic plates and use integral transforms and a collocation method to form a line-spring model for a cracked plate. The eigth-order model allows satisfaction of the three standard plate bending boundary conditions; the normal moment, twisting moment, and transverse shear force, and an additional shear stress resultant that allows analysis of transverse normal stresses near the crack tip. The line-spring model is used to develop geometry correction factors for bending of finite-thickness plates, accounting for transverse shear deformation and pressurization of the plate near the crack tip. The line-spring model is then applied to the problem of a plate with a reinforced crack, and the results are used to validate an interpolation solution based on an energy method. While not explicitly analysed, the models are applicable to many problems, including bending of bonded repairs, fracture and fatigue of composite and layered materials, surface cracks, crack tip plasticity and crack closure or crack face interaction.     

SMA-based smart soft composite structure capable of multiple modes of actuation

This paper introduces the design of a smart soft composite (SSC) actuator capable of multiple modes of actuation. This actuator combines four shape memory alloy (SMA) wires embedded in a soft matrix where one or two SMA wires can be activated to induce the actuator into either the bending mode, the twisting mode or the combined bending and twisting mode of actuation. Experimental results for actuators of different lengths were obtained for all modes of actuation and the actuator is capable of large deformations in all modes and directions of actuation. Then a simple FEA model was used to predict the range of deformation for different lengths in the different modes of actuation. This model is able to predict accurately the bending and twisting angles of the actuator for the different modes of actuation. The 120 mm actuator is capable of deformations up to approximately 160° in both the pure bending and pure twisting modes and of approximately 80° for both twisting and bending in the combined twisting and bending mode of actuation.     

Solution of one mixed problem of plate bending for a domain with partially unknown boundary

The paper addresses the problem of finding a full-strength contour in the problem of plate bending for a cycle-symmetric doubly connected domain. An isotropic elastic plate, bounded by a regular polygon, is weakened by a required full-strength hole whose symmetry axes are the regular polygon diagonals. Rigid bars are attached to each component of the broken line of the outer boundary of the plate. The plate bends under the action of concentrated moments applied to the middle points of the bars. An unknown part of the boundary is free from external forces. Using the methods of complex analysis, the analytical image of Kolosov?CMuskhelishvili??s complex potentials (characterising an elastic equilibrium of the body) and of an unknown full-strength contour are determined. A numerical analysis is performed and the corresponding plots are obtained by means of the Mathcad system.     

Moment Distribution around Circular/Elliptical/Triangular Cutouts in Infinite Symmetric Laminated Plate

In this article, a general solution for moment distribution around a circular/elliptical/triangular shaped hole in a laminated composite plate subjected to a bending/twisting moment at infinity is presented. The numerical results are obtained for graphite/epoxy, glass/epoxy, boron/epoxy, etc. The effect of loading factor, stacking sequence, material parameters, and hole geometry on moment distribution is studied for cross ply and angle ply laminates. The results are compared with existing literature and found to be in good agreement.